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\title{Probability}
\author{Last Updated: June 1977}
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In the following syllabus, the candidate should be able to state the theorems in reasonable generality and to prove a significant special case. The candidate should be acquainted with examples illustrating the theorems and the basic concepts.

\paragraph{Foundations:}
The measure theory necessary to define a probability space appropriate
for the analysis of a sequence $X_1, X_2, \dots$, of random variables,
in particular Kolmogorov's extension theorem. The definition and basic
properties of conditional probability and conditional expectation with
respect to events, random variables, and sigma fields. The various
types of convergence for a sequence of random variables; convergence
with probability one, in the mean, in probability and in distribution.
The standard distributions, \textit{e.g.}, normal, binomial,
exponential. The definition and basic properties of generating
functions and characteristic functions and their role in studying
convergence of random variables.

\paragraph{Sums of Independent Random Variables.}
The law of large numbers and the central limit theory. The definition
of the stable laws and their relation to limit laws. The renewal
theorem and its applications to, for example, Poisson processes and
queues.

\paragraph{Other Stochastic Processes.}
The classification and basic limit theorems for discrete time
denumerable state Markov chains. The definition of birth and death
processes and Brownian motion and a description of the sample
functions of these processes. The definition of a stationary process;
the ergodic theorem. The definition of martingales; the martingale
convergence theorem.

\subsection*{REFERENCES}

\begin{enumerate}
\item Lamperti, \textit{Probability}.
\item Feller, \textit{Probability Theory and its Applications, Volumes
    I and II}.
\item Chung, \textit{A Course in Probability Theory}.
\item Breiman, \textit{Probability}.
\item Dynkin and Uyshkevich, \textit{Markov Processes: Theorems and
    Problems}.
\item Neveu, \textit{Mathematical Foundations of the Calculus of
    Probability}.
\end{enumerate}

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